Derivative Continuity of the Basis Function at its Ends

We look first at the original question What is the level of derivative continuity at the ends of the basis function . 

This can be addressed informally by looking at the rate at which the final entry of the mask reduces, compared with the rate at which the gap between the last control and the end of the support reduces. 

If the final non-zero entry in the mask has the value / and the arity is a, yj is the final non-zero in the y-times refined polygon (so that / = yo) and Xj is the distance from that entry to the end of the support then yj = yq and Xj = s(a i), where s is half the support width, the initial value of x (so that s = x0). 

The right hand side is constant, independent of j, and so the left hand side must be also. For this to be the case we must have yo ak = 1, or yo = k Thus k = -logo( y0 ) = -log(M)/log(a) = -loga( / ) 

Then we expect derivatives lower than the k th to be continuous at the end of the support region, and the next derivative to have a discontinuity. 

---